RETRACTED: Non-Hermiticity of tunneling time in a spacetime-symmetric extension of nonrelativistic quantum mechanics

Abstract

Time continues to be an intriguing physical property in the modern era. On the one hand, we have the classical and relativistic notions of time, where space and time have the same hierarchy, which is essential in describing events in spacetime. On the other hand, in quantum mechanics time appears as a classical parameter, meaning that it does not have an uncertainty relation with its canonical conjugate. In this work, we use a recent spacetime-symmetric proposal [Phys. Rev. A 95, 032133 (2017)] that tries to solve the unbalance in nonrelativistic quantum mechanics by extending the usual Hilbert space, having the time parameter $t$ and the position operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{X}$ in one subspace and the position parameter $x$ and time operator $\mathbb{T}$ in the other subspace. Time as an operator is better suitable for describing tunneling processes. We then solve the $1/2$-fractional integrodifferential equation for a particle subjected to strong and weak potential limits and obtain an analytical expression for the tunneling time through a rectangular barrier. Using a Gaussian energy distribution, we demonstrate that, for wavepackets well resolved in time, the expectation value of the operator $\mathbb{T}$ is the energy average of the classical time ${T}_{\text{class}}=\ensuremath{\partial}S/\ensuremath{\partial}E$, where $S$ is the classical action, which can be real or imaginary. For wavepackets not well resolved in time, the contribution of ${T}_{\text{class}}$ consistently vanishes, and solely properties of the energy distribution contribute to $\mathbb{T}$. We show that the time of travel for nontunneling particles is purely real. When tunneling is involved, complex arrival times emerge, becoming a signature of tunneling. Furthermore, we apply our results to a constant energy distribution, obtaining pure imaginary times for energies below the barrier while obtaining complex times for particles with a wavepacket spreading energies below and above the barrier, and show a comparison to previous works.